Question

Two circles intersect each other at the point A and B. A point P and the
circumference of one circle is joined with A and B and extended to meet
the other circle at points C and D respectively. Prove that CD is parallel
to the tangent at P to the circle on which P lies.​

Answer 1

ans

Join AB and let XY be the tangent at P. Then by alternate segment theorem,

∠APX=∠ABP ……………(i)

Next, ABCD is a cyclic quadrilateral, therefore, by the theorem sum of the opposite angles of a quadrilateral is 180^{\circ}

∠ABD+∠ACD=180

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Also, ∠ABD=∠ABP=180

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(Linear Pair)

∴∠ACD=∠ABP ...........(ii)

From (i) and (ii),

∠ACD=∠APX

∴XY∥CD (Since alternate angles are equal).